In [2] the notion of “uniformly ideal” was introduced and developed the basic theory. In this article we introduce and advance a theory which, in a sense, dual to that i.e, the notion of “uniformly secondary module”.
Mohammad Bagherpoor, Abdoljavad Taherizadeh, Volume 9, Issue 4 (12-2023)
Abstract
All rings throughout this paper are commutative and Noetherian. Semidualizing modules were studied independently by Foxby [4], Golod [5], and Vasconcelos [11]. A finite R-module C is called semidualizing if the natural homothety map is an isomorphism and for all . If a semidualizing R-module has finite injective dimension, it is called dualizing and is denoted by D. The ring itself is an example of a semidualizing R-module. Many researchers, in particular Sather-Wagstaff [12], have studied the semidualizing modules. Let M be an R-module. The trace ideal of M, denoted by , is the sum of images of all homomorphisms from M to R. Trace ideals have attracted the attention of many researchers in recent years. In particular, Herzog et al. [6] and Dao et al. [3] studied the trace ideals of canonical modules. Also, the trace ideals of semidualizing modules were studied in [1]. In this paper, we study the trace ideals of tensor product of two arbitrary modules. We prove some known facts with a different approach via trace ideals. For example, let C and be two semidualizing R-modules. We show that is projective if and only if C and are projective R-modules of rank 1. Also, we study the trace ideals of maximal Cohen-Macaulay modules over a Gorenstein local ring.